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Optimized equations for X1(N)
The table below gives links to optimized equations f(x,y)=0 for X1(N), together with parametizations
E=[a1(x,y),a2(x,y),a3(x,y),a4(x,y),a6(x,y)] and
P=[u(x,y),v(x,y)]
that define an elliptic curve
v2 + a1uv + a3v = u3 + a2u2 + a4u + a6
in Weierstrass form on which P is a point of order N.
This extends results for N ≤ 50 described in Constructing elliptic curves with prescribed torsion over finite fields to N ≤ 100, and for many N ≤ 50 gives equations of lower degree (for N > 100, see this table).
In each case the equations below give a map to P1 (the function y) that matches the upper bound on the gonality of X1(N) given by Derrickx and van Hoeij
in Gonality of the modular curve X1(N). These bounds are known to be tight for N ≤ 40.
This is joint work in progress with Mark van Hoeij (last updated January 10, 2014). Partially funded by NSF grant DMS-1115455.
| N | degx | degy | f(x,y)
| N | degx | degy | f(x,y)
| N | degx | degy | f(x,y)
| N | degx | degy | f(x,y)
| N | degx | degy | f(x,y)
|
| 6 | 0 | 1 | f(x,y)
| 25 | 8 | 5 | f(x,y)
| 44 | 22 | 15 | f(x,y)
| 63 | 72 | 36 | f(x,y)
| 82 | 63 | 58 | f(x,y)
|
| 7 | 0 | 1 | f(x,y)
| 26 | 7 | 6 | f(x,y)
| 45 | 36 | 18 | f(x,y)
| 64 | 48 | 32 | f(x,y)
| 83 | 95 | 90 | f(x,y)
|
| 8 | 0 | 1 | f(x,y)
| 27 | 8 | 6 | f(x,y)
| 46 | 21 | 19 | f(x,y)
| 65 | 53 | 42 | f(x,y)
| 84 | 72 | 48 | f(x,y)
|
| 9 | 0 | 1 | f(x,y)
| 28 | 9 | 6 | f(x,y)
| 47 | 30 | 29 | f(x,y)
| 66 | 60 | 30 | f(x,y)
| 85 | 90 | 72 | f(x,y)
|
| 10 | 0 | 1 | f(x,y)
| 29 | 12 | 11 | f(x,y)
| 48 | 24 | 16 | f(x,y)
| 67 | 59 | 58 | f(x,y)
| 86 | 73 | 64 | f(x,y)
|
| 11 | 2 | 2 | f(x,y)
| 30 | 8 | 6 | f(x,y)
| 49 | 31 | 21 | f(x,y)
| 68 | 54 | 36 | f(x,y)
| 87 | 140 | 70 | f(x,y)
|
| 12 | 0 | 1 | f(x,y)
| 31 | 12 | 12 | f(x,y)
| 50 | 23 | 15 | f(x,y)
| 69 | 88 | 44 | f(x,y)
| 88 | 90 | 60 | f(x,y)
|
| 13 | 3 | 2 | f(x,y)
| 32 | 12 | 8 | f(x,y)
| 51 | 48 | 24 | f(x,y)
| 70 | 45 | 36 | f(x,y)
| 89 | 108 | 104 | f(x,y)
|
| 14 | 2 | 2 | f(x,y)
| 33 | 20 | 10 | f(x,y)
| 52 | 31 | 21 | f(x,y)
| 71 | 69 | 66 | f(x,y)
| 90 | 67 | 48 | f(x,y)
|
| 15 | 2 | 2 | f(x,y)
| 34 | 10 | 10 | f(x,y)
| 53 | 39 | 37 | f(x,y)
| 72 | 45 | 32 | f(x,y)
| 91 | 106 | 84 | f(x,y)
|
| 16 | 3 | 2 | f(x,y)
| 35 | 15 | 12 | f(x,y)
| 54 | 25 | 18 | f(x,y)
| 73 | 73 | 70 | f(x,y)
| 92 | 99 | 66 | f(x,y)
|
| 17 | 4 | 4 | f(x,y)
| 36 | 11 | 8 | f(x,y)
| 55 | 37 | 30 | f(x,y)
| 74 | 52 | 51 | f(x,y)
| 93 | 160 | 80 | f(x,y)
|
| 18 | 3 | 2 | f(x,y)
| 37 | 19 | 18 | f(x,y)
| 56 | 36 | 24 | f(x,y)
| 75 | 63 | 40 | f(x,y)
| 94 | 83 | 83 | f(x,y)
|
| 19 | 5 | 5 | f(x,y)
| 38 | 13 | 12 | f(x,y)
| 57 | 60 | 30 | f(x,y)
| 76 | 67 | 45 | f(x,y)
| 95 | 113 | 90 | f(x,y)
|
| 20 | 3 | 3 | f(x,y)
| 39 | 28 | 14 | f(x,y)
| 58 | 32 | 31 | f(x,y)
| 77 | 74 | 60 | f(x,y)
| 96 | 81 | 56 | f(x,y)
|
| 21 | 5 | 4 | f(x,y)
| 40 | 18 | 12 | f(x,y)
| 59 | 48 | 46 | f(x,y)
| 78 | 84 | 42 | f(x,y)
| 97 | 129 | 123 | f(x,y)
|
| 22 | 5 | 4 | f(x,y)
| 41 | 23 | 22 | f(x,y)
| 60 | 35 | 24 | f(x,y)
| 79 | 85 | 82 | f(x,y)
| 98 | 92 | 63 | f(x,y)
|
| 23 | 7 | 7 | f(x,y)
| 42 | 24 | 12 | f(x,y)
| 61 | 51 | 49 | f(x,y)
| 80 | 72 | 48 | f(x,y)
| 99 | 180 | 90 | f(x,y)
|
| 24 | 5 | 4 | f(x,y)
| 43 | 26 | 24 | f(x,y)
| 62 | 36 | 36 | f(x,y)
| 81 | 77 | 54 | f(x,y)
| 100 | 95 | 60 | f(x,y)
|